Critical currents in superconductors with quasiperiodic pinning arrays:
One-dimensional chains and two-dimensional Penrose lattices
arXiv:cond-mat/0511098v1 [cond-mat.supr-con] 4 Nov 2005
1
Vyacheslav Misko1,2 , Sergey Savel’ev1 , and Franco Nori1,2
Frontier Research System, The Institute of Physical and Chemical
Research (RIKEN), Wako-shi, Saitama, 351-0198, Japan and
2
Center for Theoretical Physics, Center for the Study of Complex Systems,
Department of Physics, University of Michigan, Ann Arbor, MI 48109-1040, USA
(Dated: February 2, 2008)
We study the critical depinning current Jc , as a function of the applied magnetic flux Φ, for
quasiperiodic (QP) pinning arrays, including one-dimensional (1D) chains and two-dimensional (2D)
arrays of pinning centers placed on the nodes of a five-fold Penrose lattice. In 1D QP chains
of pinning sites, the peaks in Jc (Φ) are shown to be determined by a sequence of harmonics of
long and short periods of the chain. This sequence includes as a subset the sequence of successive
Fibonacci numbers. We also analyze the evolution of Jc (Φ) while a continuous transition occurs from
a periodic lattice of pinning centers to a QP one; the continuous transition is achieved by varying
the ratio γ = aS /aL of lengths of the short aS and the long aL segments, starting from γ = 1 for a
periodic sequence. We find that the peaks related to the Fibonacci sequence are most pronounced
when γ is equal to the “golden mean”. The critical current Jc (Φ) in QP lattice has a remarkable
self-similarity. This effect is demonstrated both in real space and in reciprocal k-space. In 2D
QP pinning arrays (e.g., Penrose lattices), the pinning of vortices is related to matching conditions
between the vortex lattice and the QP lattice of pinning centers. Although more subtle to analyze
than in 1D pinning chains, the structure in Jc (Φ) is determined by the presence of two different kinds
of elements forming the 2D QP lattice. Indeed, we predict analytically and numerically the main
features of Jc (Φ) for Penrose lattices. Comparing the Jc ’s for QP (Penrose), periodic (triangular)
and random arrays of pinning sites, we have found that the QP lattice provides an unusually broad
critical current Jc (Φ), that could be useful for practical applications demanding high Jc ’s over a
wide range of fields.
PACS numbers: 74.25.Qt
I.
INTRODUCTION
Recent progress in the fabrication of nanostructures
has provided a wide variety of well-controlled vortexconfinement topologies, including different types of regular pinning arrays. A main fundamental question in this
field is how to drastically increase vortex pinning, and
thus the critical current Jc , using artificially-produced
periodic arrays of pinning sites (APS). These periodic
APS have been extensively used for studying vortex pinning and vortex dynamics. In particular, enhanced Jc
and commensurability effects have been demonstrated in
superconducting thin films with square and triangular
arrays of sub-µm holes (i.e., antidots) [1, 2, 3, 4, 5, 6].
Moreover, blind antidots (i.e., holes which partially perforate the film to a certain depth) [7], or pinning arrays with field-dependent pinning strength [8], provide
more flexibility for controlling properties such as pinning
strength, anisotropy, etc. The increase and, more generally, control of the critical current Jc in superconductors
by its patterning (perforation) can be of practical importance for applications in micro- and nanoelectronic
devices.
A peak in the critical current Jc (Φ), for a given value
of the magnetic flux, say Φ1 , can be engineered using
a superconducting sample with a periodic APS with a
matching field H1 = Φ1 /A (where A is the area of the
pinning cell), corresponding to one trapped vortex per
pinning site. However, this peak in Jc (Φ), while useful
to obtain, decreases very quickly for fluxes away from Φ1 .
Thus, the desired peak in Jc (Φ) is too narrow and not
very robust against changes in Φ. It would be greatly
desirable to have samples with APS with many periods.
This multiple-period APS sample would provide either
very many peaks or an extremely broad peak in Jc (Φ),
as opposed to just one (narrow) main peak (and its harmonics). We achieve this goal (a very broad Jc (Φ)) here
by studying samples with many built-in periods.
The development of new fabrication technologies for
pinning arrays with controllable parameters allows to
fabricate not only periodic (square or triangular) but also
more complicated quasiperiodic (QP) arrays of pinning
sites, including Penrose lattices [9, 10, 11].
The investigation of physical properties of QP systems
has attracted considerable interest including issues such
as band structure and localization of electronic states in
two-dimensional (2D) Penrose lattice [12, 13], electronic
and acoustic properties of one-dimensional (1D) QP lattices [14, 15], superconducting-to-normal phase boundaries of 2D QP micronetworks [16, 17, 18, 19, 20], QP
semiconductor heterostructures and optical superlattices
[21], soliton pinning by long-range order [22] and pulse
propagation [23] in QP systems. Moreover, increasing
and, more generally, controlling the critical current in
superconductors by its patterning (perforation) can be
of practical importance for applications in micro- and
2
nanoelectronic devices.
The original tiling has been studied in Ref. [9]. The inflation (or production) rules of “finite Penrose patterns”
generated by repeated application of deflation and rescaling have been found, which show a definite hierarchical
structure of the Penrose patterns [11].
The electronic and acoustic properties of a onedimensional quasicrystal have been studied in Refs. [13,
14]. It has been shown, in particular, that there exist two
types of the wave functions, self-similar (fractal) and nonself-similar (chaotic) which show “critical” or “exotic”
behavior [13]. By both numerical (non-perturbative) and
analytical (perturbative) approaches, it has been demonstrated [14, 15] that the phonon and electronic spectra of
1D quasicrystals exhibit a self-similar hierarchy of gaps
and localized states in the gaps. The existence of gaps,
and gap states, in QP GaAs-AlGaAs superlattices has
been predicted and found experimentally.
Along with studying the structural, electronic and
acoustic properties of QP structures, considerable
progress has been reached in understanding the superconducting properties of 2D quasicrystalline arrays
[16, 17, 18, 19, 20]. The effect of frustration, induced
by a magnetic field, on the superconducting diamagnetic
properties has been revealed and the superconductingto-normal phase boundaries, Tc (H), have been calculated for several geometries with quasicrystalline order,
in a good agreement with experimentally measured phase
boundaries [17, 19]. A comprehensive analysis of superconducting wire networks including quasicrystalline geometries and Josephson-junction arrays in a magnetic
field has been presented in Refs [19, 20]. An analytical
approach [19, 20] was introduced to analyze the structures which are present in phase diagrams for a number
of geometries. It has been shown that the gross structure
is determined by the statistical distributions of the cell
areas, and that the fine structures are determined by correlations among neighboring cells in the lattices. The effect of thermal fluctuations on the structure of the phase
diagram has been studied [19] by a cluster mean-field
calculation and using real-space renormalization group.
In this paper, we study another phenomenon related
to superconducting properties of quasiperiodic systems,
namely, vortex pinning by 1D QP chains and by 2D arrays of pinning sites located at the nodes of QP lattices
(e.g., a five-fold Penrose lattice). It should be noted that
in superconducting networks the areas of the network plaquettes play a dominant role [16, 17, 18, 19]. However,
for vortex pinning by QP pinning arrays, the specific geometry of the elements which form the QP lattice and
their arrangement (and not just the areas) are important, making the problem complicated.
In Sec. II we introduce the model used for describing
vortex dynamics in QP pinning arrays and for determining the critical depinning current, Jc , which is analyzed
for different quasicrystalline geometries.
The pinning of vortices by a 1D QP chain of pinning
sites (i.e., Fibonacci sequence) is discussed in Sec. III. We
consider a continuous transition from a periodic to the
QP chain of pinning sites and we monitor the corresponding changes in the critical current Jc as a function of the
applied magnetic flux Φ. A remarkable self-similarity of
Jc (Φ) is demonstrated in both real space and in reciprocal k-space.
Section IV studies the pinning of flux lattices by 2D
QP pinning arrays including the five-fold Penrose lattice. We analyze changes of the function Jc (Φ) during
a continuous transition from a periodic triangular lattice of pinning sites to the Penrose lattice. Based on
detailed considerations of the structure and specific local
rules of construction of the Penrose lattice, we predict
the main features of the function Jc (Φ). Numerical simulations with different finite-size Penrose lattices confirm
the predicted main features for large-size lattices, that is
important for possible experimental observations of the
revealed quasiperiodic features. We also obtain analytical results supporting our conclusions. Moreover, we
also discuss the changes in the critical current by adding
either a “quasiperiodic” modulation or random displacements to initially periodic pinning arrays.
II.
MODEL
We model a three-dimensional (3D) slab infinitely long
in the z-direction, by a two-dimensional (2D) (in the xyplane) simulation cell with periodic boundary conditions,
assuming the vortex lines are parallel to the cell edges.
To study the dynamics of moving vortices driven by a
Lorentz force, interacting with each other and with pinning centers, we perform simulated annealing simulations
by numerically integrating the overdamped equations of
motion (see, e.g., Ref. [24, 25, 26, 27]):
ηvi = fi = fivv + fivp + fiT + fid .
(1)
Here, fi is the total force per unit length acting on vortex
i, fivv and fivp are the forces due to vortex-vortex and
vortex-pin interactions, respectively, fiT is the thermal
stochastic force, and fid is the driving force acting on the
i-th vortex; η is the viscosity, which is set to unity. The
force due to the interaction of the i-th vortex with other
vortices is
Nv
X
| ri − rj |
r̂ij ,
(2)
f0 K 1
fivv =
λ
j
where Nv is the number of vortices, K1 is a modified
Bessel function, λ is the magnetic field penetration depth,
r̂ij = (ri − rj )/ | ri − rj |, and
f0 =
Φ20
.
8π 2 λ3
Here Φ0 = hc/2e is the magnetic flux quantum. It is
convenient, following the notation used in Ref. [24, 25,
26, 27], to express all the lengths in units of λ and all
3
the fields in units of Φ0 /λ2 . The Bessel function K1 (r)
decays exponentially for r greater than λ, therefore it is
safe to cut off the (negligible) force for distances greater
than 5λ. The logarithmic divergence of the vortex-vortex
interaction forces for r → 0 is eliminated by using a cutoff
for distances less than 0.1λ.
Vortex pinning is modeled by short-range parabolic po(p)
tential wells located at positions rk . The pinning force
is
!
Np (p)
X
rp − | ri − rk |
fp
(p)
(p)
vp
fi =
r̂ik ,
| ri − rk | Θ
rp
λ
k
(3)
where Np is the number of pinning sites, fp is the maximum pinning force of each potential well, rp is the range
of the pinning potential, Θ is the Heaviside step function,
(p)
(p)
(p)
and r̂ik = (ri − rk )/ | ri − rk | .
The temperature contribution to Eq. (1) is represented
by a stochastic term obeying the following conditions:
hfiT (t)i = 0
(4)
hfiT (t)fjT (t′ )i = 2 η kB T δij δ(t − t′ ).
(5)
and
The ground state of a system of moving vortices is obtained as follows. First, we set a high value for the temperature, to let vortices move randomly. Then, the temperature is gradually decreased down to T = 0. When
cooling down, vortices interacting with each other and
with the pinning sites adjust themselves to minimize the
energy, simulating the field-cooled experiments [29, 30].
In order to find the critical depinning current, Jc , we
apply an external driving force gradually increasing from
fd = 0 up to a certain value fd = fdc , at which all the
vortices become depinned and start to freely move. For
values of the driving force just above fdc , the total current of moving vortices J ∼ hvi becomes nonzero. Here,
hvi is the normalized (per vortex) average velocity of all
the vortices moving in the direction of the applied driving
force. In numerical simulations, this means that, in practice, one should define some threshold value Jmin larger
than the noise level. Values larger than Jmin are then considered as nonzero currents. However, instead using this
criterion-sensitive scheme, we can use an alternative approach based on potential energy considerations. In case
of deep short-range potential wells, the energy required
to depin vortices trapped by pinning sites is proportional
(p)
to the number of pinned vortices, Nv . Therefore, in
this approximation we can define the “critical current”
as follows:
(p)
Jc (Φ) = J0
Nv (Φ)
,
Nv (Φ)
(6)
where J0 is a constant, and study the dimensionless value
Jc′ = Jc /J0 (further on, the primes will be omitted).
Throughout this work, we use narrow potential wells as
pinning sites, characterized by rp = 0.04λ to 0.1λ. Our
calculations show that, for the parameters used, the dependence of the critical current Jc (Φ) defined according to Eq. (6), is in good agreement to that based on
the above general definition of Jc (which involves an adjustable parameter Jmin ). The advantages of using Jc
defined by Eq. (6) are the following: it (i) does not involve any arbitrary threshold Jmin and (ii) is less CPUtime consuming, allowing the study of very large-size lattices. Moreover, the goal of this study is to reveal specific matching effects between a (periodic) vortex lattice
and arrays of QP pinning sites, and to study how the
quasiperiodicity manifests itself in experimentally mea(p)
surable quantities (Jc , Nv /Nv ) related to the vortex
pinning by QP (e.g., the Penrose lattice) pinning arrays.
III. PINNING OF VORTICES BY A 1D
QUASIPERIODIC CHAIN OF PINNING SITES
In this section we study the pinning of vortices by one
dimensional (1D) QP chains of pinning sites.
A.
1D quasiperiodic chain
As an example of a 1D QP chain, or 1D quasicrystal,
a Fibonacci sequence is considered, which can be constructed following a simple procedure: let us consider
two line segments, long and short, denoted, respectively,
by L and S. If we place them one by one, we obtain an
infinite periodic sequence:
LSLSLSLSLSLSLSLS ...
(7)
A unit cell of this sequence consists of two elements, L
and S. In order to obtain a QP sequence, these elements
are transformed according to Fibonacci rules as follows:
L is replaced by LS, S is replaced by L:
L → LS,
S → L.
(8)
As a result, we obtain a new sequence:
LSLLSLLSLLSLLSL ...
(9)
Iteratively applying the rule (8) to the sequence (9),
we obtain, in the next iteration, a sequence with a fiveelement unit cell (LSLLS), then with an eight-element
unit cell (LSLLSLSL), and so on, to infinity. For the
sequence with an n-element cell, where n → ∞, the ratio
of numbers of long to short elements is the golden mean
value,
√
τ = (1 + 5)/2.
(10)
The position of the nth point where a new element, either
L or S, begins is determined by [11]:
xn = n + γ [γn] ,
(11)
4
B.
(a)
Pinning of vortices by a 1D periodic chain of
pinning sites
rp = 0.1l
Jc
0.8
0.4
0.0
(b)
rp = 0.04l
Jc
0.8
0.4
0.0
0
F1
F2
F3
F
FIG. 1: (a, b) Dimensionless critical depinning current Jc ,
as a function of the applied magnetic flux Φ, in a 1D periodic
chain of pinning sites, for a cell containing 25 pinning sites,
Np = 25 and fp /f0 = 2.0. The indicated fluxes Φ1 , Φ2 ,
and Φ3 correspond to the first, second, and third matching
fields. The function Jc (Φ) is shown for two different values
of the pinning site radius, rp = 0.1λ (a), and rp = 0.04λ
(b). When rp , decreases, the main commensurability peaks
become sharper, as shown in (a) and (b).
where [x] denotes the maximum integer less or equal to
x. Equation (11) corresponds to the case when the Fibonacci sequence has a ratio γ = aS /aL of the length of
the short segment, aS , to the length of the long segment,
aL . Ratios γ other than 1/τ correspond to other chains
which are all QP. Along with the golden mean value of
γ = 1/τ , we use in our simulations γ’s varying in the
interval between 0 and 1: 0 < γ < 1, when analyzing a
continuous transition from a periodic to a QP (Fibonacci
sequence) pinning array.
To study the critical depinning current Jc in 1D QP
pinning chains, we place pinning sites on the points where
L or S elements of the QP sequence link to each other.
Therefore, the coordinates of the centers of the pinning
sites are defined by Eq. (11) with γ = aS /aL .
We start with a periodic chain of pinning sites, which
can be considered, in the framework of the above scheme,
as a limiting case of a “QP” chain with γ = 1, i.e. aS =
(p)
aL = 1. In Fig. 1a, the critical current Jc ∼ Nv /Nv
is shown as a function of the applied magnetic flux Φ
for fp /f0 = 2 and for a pinning site radius rp = 0.1λ.
Sharp peaks of the function Jc (Φ) correspond to matching fields. Since the dimensionless critical current we
plot is that per vortex and is proportional to the number of pinned vortices and inversely proportional to the
total number of vortices (Eq (6)), therefore, the magnitude of the peaks versus Φ ∼ Nv decreases as 1/Φ.
Then the maximum heights of the peaks are: Jc (Φ1 ) = 1,
Jc (2Φ1 ) = 0.5, and Jc (3Φ1 ) = 0.33. Note that these values are obtained provided each pinning site can trap only
one vortex, which is justified for the chosen radius of the
pinning site and for the vortex densities considered. In
addition, there are weak wide maxima corresponding to
Φ1 /2 and other “subharmonics”, i.e. Φi + Φ1 /2 (Fig. 1a),
where i is an integer. For very small values of Φ1 , the
vortex density is very low, and vortices almost do not
interact with each other. As a result, in the ground state
they all become trapped by pinning sites, and Jc is maximal for small Φ1 .
For smaller radii of the pinning sites, rp = 0.04λ
(Fig. 1b), the Jc (Φ) peaks corresponding to matching
fields become sharper because for smaller values of rp ,
it is more difficult to fulfill the commensurability conditions. Any features around the main peaks, including
subharmonics, are suppressed. Note also a “parity effect”
takes place in this case: since the number of pinning sites
per cell is odd (Np = 25 in Fig. 1), therefore Jc (Φ) peaks
are suppressed for Φ1 /2 and for even values of i in the
sequence Φi + Φ1 /2.
IV. PINNING OF VORTICES BY A 1D
QUASIPERIODIC CHAIN OF PINNING SITES:
GRADUAL EVOLUTION FROM A PERIODIC
TO A QUASIPERIODIC CHAIN
Let us consider a QP chain of pinning sites with spacings between pinning sites given by aL (long) and aS
(short). The long and short segments alternate according to the Fibonacci rules forming a Fibonacci sequence.
The number of pinning sites per cell coincides with the
number of elements of this sequence per unit cell. It is
natural to take a chain with a number of elements equal
to one of the successive Fibonacci numbers as a 1D cell,
although in principle it could be of any length. We impose periodic boundary conditions at the ends of the cell.
The larger cell we take, the closer we are to describing a truly QP structure. However, it turns out, that
even a finite part of a QP system (1D chain or 2D QP
lattice) provides us with reliable information concerning
5
g = 1.0
Critical current Jc (arbitrary units) versus number of vortices
g = 0.9
g = 0.8
g = 0.7
g = 1/t = 0.618
g = 0.6
g = 0.5
g = 0.4
g = 0.3
g = 0.2
g = 0.1
0
42
13 21
34
63
55
84
89
105
Nv
FIG. 2: (Color) Critical depinning current Jc , as a function of the applied magnetic flux (for convenience, shown as
a function of the number of vortices, Nv ∼ Φ), in a 1D QP
chain of pinning sites with fp /f0 = 1, rp = 0.1λ, for chains
characterized by different ratios of the lengths of short aS to
long aL spacings, γ = aS /aL . For small deviations of the
chain from a periodic chain (γ = 0.9 to 1), commensurability
peaks are similar to those shown in Fig. 1. For intermediate values of γ (γ = 0.2 to 0.8), peaks are determined by
a sequence of harmonics of numbers of long and short periods of the chain, which includes the sequence of successive
Fibonacci numbers,
√ most pronounced for γ = 1/τ ≈ 0.618,
where τ = (1 + 5)/2 ≈ 1.618 is the golden mean. For very
small γ (e.g., γ = 0.1), the QP chain effectively becomes periodic but with the number of pinning sites equal to the number
of long periods in the chain.
properties of the whole system. This is based on the
structural self-similarity of QP systems. These properties are studied here for the critical depinning current
Jc and have also been demonstrated for other physical
phenomena [14, 15, 17, 18, 19, 20, 28].
Figure 2 shows the evolution of Jc as a function of the
number of vortices, Nv ∼ Φ, for various values of the
parameter γ. The top curve represents the limiting case
of a periodic chain (γ = 1.0) with typical peak structure
discussed above (Fig. 1). The chain contains Np = 21
pinning sites. As a result, commensurability peaks appear at Nv = 21, 42, 63, etc., i.e. multiples of 21. A small
QP distortion of the chain does not appreciably affect the
peak structure of the function Jc (Nv ) (γ = 0.9). When
the deviation of the factor γ from unity becomes larger
(γ = 0.8, 0.7), commensurability peaks for Nv = 42, 63,
and other multiples of 21 decrease in magnitude. At the
same time, new peaks appear at Nv = 55, 76 (= 55 + 21),
97 (= 76 + 21) for γ = 0.8. Then, with further decrease
of γ (γ = 0.7), these peaks remain (Nv = 55 even grows
in magnitude), and a new peak at Nv = 34 arises.
For the golden mean-related value of γ = 1/τ ≈ 0.618,
we obtain a set of peaks, which are “harmonics” of the
numbers of long and short periods of the chain (or reciprocal lengths aL and aS ), i.e.
xQP
peaks,
i
= Ai NL + Bi NS =
A′i
B′
+ i,
aS
aL
(12)
where NL and NS are the numbers of long and short elements, respectively; Ai (A′i ) and Bi (Bi′ ) are generally
(positive or negative) multiples or divisors of NL and
NS ; the upper index “QP” denotes “quasiperiodic”. It is
easy to see that this set includes as a subset the sequence
of successive Fibonacci numbers. In particular, the following well-resolved peaks of the function Jc (Nv ) appear
for γ = 1/τ ≈ 0.618: Nv = 13 (= NL = 13, 13 is a Fibonacci number (FN)); Nv = 17 (= (2NL +NS )/2, where
(2NL + NS ) = 34 is a FN); Nv = 21 (= Np = NL + N S,
21 is a FN); Nv ≈ 27-28 (= (3NL + 2NS )/2, where
(3NL + 2NS ) = 55 is a FN); Nv = 34 (2NL + NS );
Nv ≈ 44-45 (= (5NL + 3NS )/2, where (5NL + 3NS ) = 89
6
(a)
g = 1/t = 0.618
Np = 21
Np = 34
Np = 55
Np = 89
Jc ~ Nv(p)/Nv (arbitrary units)
FIG. 3: (Color) (a) The critical depinning current Jc , as a
function of the number of vortices, Nv ∼ Φ, for different 1D
QP chains, Np = 21 (red bottom line), Np = 34 (blue line),
Np = 55 (green line), and Np = 89 (dark blue top line), and
the same γ = aS /aL = 1/τ . The parameters used here are:
fp /f0 = 1.0 and rp = 0.1λ. Independently of the length of
the chain, the peaks for all of the curves include as a subset
the sequence of successive Fibonacci numbers (indicated by
blue arrows in the horizontal axis) and their sub-harmonics.
(b) Jc (Nv ) for a long chain Np = 144 and the same γ =
1/τ . Notice that now Nv is much larger than in (a). (c) The
function Jc (Φ/Φ1 ) for the same set of 1D chains (using the
same colors), normalized by the number of pinning sites for
each chain. The curves for chains with different Np ’s display
the same sets of peaks, namely, at Φ/Φ1 = 1 (first matching
field) and Φ/Φ1 = 0.5, as well as at the golden-mean-related
values: Φ/Φ1 = τ /2, Φ/Φ1 = (τ + 1)/2 = τ 2 /2, Φ/Φ1 = τ ,
Φ/Φ1 = (τ +τ 2 )/2 = τ 3 /2, Φ/Φ1 = τ 2 = τ +1, Φ/Φ1 = τ 2 +1.
Therefore, similar sets of peaks are obtained for both cases:
for the curves plotted in the same scale (a), (b); for the curves
plotted in individual scales, i.e., normalized on the number of
pinning sites in the chain (c). This behavior demonstrates the
self-similarity of Jc (Φ).
0
50
55
5 8 13 21 34
100
150
Nv
89
144
1
(b)
0
0
1
0.5
233
Np = 21
Np = 34
Np = 55
Np = 89
Np = 144
t
1
Nv
377
g = 1/t = 0.618
t/2
600
400
200
144
Jc ~ Nv(p)/Nv
The above QP peaks only slightly degrade at γ = 0.6.
However, when the length of the long segment aL becomes twice the length of the short one aS , i.e. γ = 0.5,
sharp commensurability peaks appear which are related
to the small segment of the chain with length aS = aL /2.
Namely, we obtain peaks at Nv = 34 ∼ (2NL + NS )
and at other values of Nv which are sub-harmonics of
2NL + NS : Nv = 17, 34, 68. Other peaks, in particular, those related to the Fibonacci sequence, are much
less pronounced for γ = 0.5. For γ = 0.4, the peaks
are at: Nv = 13, 17, 34, 47 (= 3NL + NS ), 68, and 81.
A very strong (recall that the maximum amplitude of
the peak is ∼ 1/Nv ) peak at Nv = 81 is a “resonance”
peak corresponding to the ratio of Bi /Ai = 2/5 = γ, i.e.
5NL + 2NS = 81. When γ = 0.3, the “resonance” ratio
Bi /Ai = 0.3, therefore, a strong peak appears for the
“nearest to 0.3” value Bi /Ai = 0.33 : 3NL + NS = 47,
and also for 6NL + 2NS = 94. Another “close to 0.3”
value is Bi /Ai = 0.25, which is responsible for the peaks
at Nv = 30 (2NL +NS /2) and Nv = 60 (4NL +NS ). Also,
peaks at Nv = 13, 17 and 55 are present. The “resonant”
peak for γ = 0.2 (i.e., for the ratio γ = Bi /Ai = 0.2),
appears at Nv = 73 (= 5NL + NS ). The closest neighboring peaks are at Nv = 60 (= 4NL + NS ) and Nv = 86
(= 6NL + NS ), [also: Nv = 43 (= 3NL + NS /2)], characterized by ratios Bi /Ai = 0.25 and Bi /Ai = 0.17, correspondingly. Finally, for γ = 0.1 we arrive at the situation
when we have an almost periodic chain but with a period
a different from that for γ = 1.0: a′ ≈ aL since aS ≪ aL .
Jc ~ Nv(p)/Nv
Np = 144
is a FN); Nv = 55 (= (3NL + 2NS ), 55 is a FN); Nv = 68
(= (4NL + 2NS )); Nv = 89 (= (5NL + 3NS ), 89 is a
FN). In summary, the most pronounced peaks are at:
Nv = 13, 17, 21, 34, 55, 89, which (except the point
Nv = 17 = 34/2) form a sequence of successive FNs.
2
t /2
(c)
t2
3
t /2
t2+1
0
0
1
2
3
F/F1
4
The number of pinning sites becomes Np = NL = 13,
and we obtain commensurability peaks at the positions
which are multiples of Np = 13, i.e., at Nv = 13, 26, 39,
52, which is typical for periodic chains of pinning sites.
In order to demonstrate that the above analysis is general and reveals the QP features independently of the
length of a specific chain of pinning sites, let us compare
results from the calculation of Jc (Nv ) for different chains.
In Fig. 3a, the function Jc (Nv ) is shown for four different
1D QP chains, Np = 21, Np = 34, Np = 55, and Np = 89,
and the same γ = 1/τ . Figure 3a clearly shows that the
7
(a)
g = 0.618
0.4
0.2
0.0
1.0
1.5
2.0
2.5
3.0
3.5
(b)
0.4
Amplitude (arbitrary units)
positions of the main peaks in Jc , i.e. those corresponding
to a Fibonacci sequence, and other peaks whose positions
are described by Eq. (12), to a significant extent, do not
depend on the length of the chain. The peaks shown in
Fig. 3 form a Fibonacci sequence: Nv = 13, 21, 34, 55, 89,
144, and other “harmonics”: Nv = 17, 27-28 (= 55/2),
44-45 (= 89/2), etc. At the same time, longer chains allow to better reveal peaks for larger Fibonacci numbers.
Thus, for chains with Np = 144 (Fig. 3b) peaks at the
next Fibonacci numbers are pronounced: Nv = 144, 233,
377.
While the curves for different chains are plotted in
Fig. 3a in the same scale, Fig. 3c shows these curves
in individual scales. Namely, we rescale each Jc by normalizing each Jc by the number of pinning sites in each
curve. Thus, Φ1 corresponds to Nv = 21 for the chain
with Np = 21, to Nv = 34 for the chain with Np = 34,
etc. After this rescaling, the Jc curves approximately follow each other and have pronounced peaks for the goldenmean-related values of Φ/Φ1 , as shown in Fig. 3c. For
example, Φ/Φ1 = τ corresponds to Nv = 34 for the chain
with Np = 21, to Nv = 55 for the chain with Np = 34, to
Nv = 89 for Np = 55, to Nv = 144 for Np = 89, and to
Nv = 233 for the chain with Np = 144. Note that these
peaks (i.e., corresponding to the golden mean) are most
pronounced for each chain in Fig. 3a,b.
Therefore, the same peaks of the function Jc (Φ) for
different chains are revealed before and after rescaling.
This means that the function Jc (Φ) for the 1D QP chain
is self-similar. Below, we demonstrate the revealed selfsimilarity effect in a reciprocal k-space.
0.2
0.0
2.6
2.4
2.2
2.0
1.8
1.6
1.4
1.2
(c)
0.4
0.2
0.0
1.8
2.0
2.2
2.4
2.6
(d)
0.4
A. Fourier-transform of the vortex distribution
function on a 1D periodic chain of pinning sites:
Self-similarity effect
As we established above, the revealed QP features
(e.g., peaks of the function Jc (Φ ∼ Nv ) ) (i) are independent of the length of the chain and (ii) the longer
chain we take, the more details (e.g., “subharmonics”) of
QP features can be observed.
This result is related to an important property of QP
systems, self-similarity, which could be better understood
by analyzing the Fourier-transform of the distribution
function of the system of vortices pinned on a QP array.
(p)
In k-space, the distribution function Fv (q) of a sys(p)
tem of Nv pinned vortices can be represented as the
inverse Fourier-transform of the 1D distribution function
(p)
Fv (n) of the vortices in real space:
Fv(p) (q)
=
1
Nv(p)
X
(p)
Nv n=1
Fv(p) (n) exp {−2πiqn/Nv(p)}. (13)
Figure 4 shows the Fourier-transform of a system of
pinned vortices for a γ equal√to the inverse golden mean:
1/γ = aL /aS = τ = (1 + 5)/2. The plots shown in
0.2
0.0
2.3
2.2
2.1
c
2.0
q 0.5 ´ 10-2 l-1
1.9
1.8
h
FIG. 4: The self-similar Fourier-transform of the distribution
function (vortex density) of the system of Nv = 144 vortices
pinned on a QP array, for γ = 1/τ . The portion limited by
the two arrows in (a) is successively magnified several times
and the corresponding results shown in (b), (c) and (d).
Figs. 4a to 4d are obtained according to the following
rule. The portion of the horizontal axis which is limited
by the two arrows in Fig. 4a is rescaled and shown in
Fig. 4b. In the same way, the portion limited by the
two arrows in Fig. 4b is rescaled and shown in Fig. 4c.
Fig. 4d is obtained following the same procedure. Note
that each subsequent scaling is accompanied by flipping
the direction of the q-axis to the opposite direction. A
similar property is clear from the experimental diffraction
patterns of quasicrystals [11]. The pentagons of Bragg
8
(a)
0.6
havior of the function Jc (Φ) when increasing the length
of the chain of pinning sites, the Fourier-transform of the
distribution function of the vortices pinned on a QP array reproduce its main features (peaks) in a self-similar
way, when increasing the range in k-space, and simultaneously acquires a more elaborate structure with smaller
self-similar peaks.
As we discussed above, the main commensurability
peaks evolve from a perfectly periodic set of the type
Φi = mΦ1 , where m > 1 is a positive integer, (through
the set of QP peaks defined by Eq. (12)), to another set
of periodic peaks Φ′i = mΦ′1 , when the “quasiperiodicity parameter” γ ≡ aS /aL is gradually tuned between
the values γ = 1.0 and γ = 0 (see Fig. 2). These limits
(γ = 1.0 and γ = 0) correspond to periodic chains.
Fig. 5 illustrates the corresponding evolution of the
Fourier-transform of the distribution function of vortices
pinned on a 1D QP array of pinning sites.
For γ = 1 (Fig. 5a), there is a single sharp peak
(accompanied by negligibly small satellites) corresponding to a periodic chain. For smaller γ’s (e.g., γ = 0.8
(Fig. 5b), γ = 0.8 (Fig. 5c), or γ = 0.618 (Fig. 4a)), a
set of satellite peaks appears around the main peak. Simultaneously, the intensity of this peak decreases giving
rise to another main peak for smaller value of q (Fig. 5d),
which corresponds to a periodic chain with a larger period.
g = 1.0
0.4
0.2
0
(b)
Amplitude
(arbitrary units)
0.6
g = 0.8
0.4
0.2
0
(c)
g = 0.5
(d)
g = 0.1
0.4
0.2
0
0.4
V. PINNING OF VORTICES BY 2D
QUASIPERIODIC PINNING ARRAYS
0.2
0
0
1
2
c
3
q 0.5 ´ 10-2 l-1
4
5
h
FIG. 5: Fourier-transform of the distribution function of the
vortices (Nv = 144) interacting with a QP array of pinning
sites, for different values of γ = 1.0 (a), 0.8 (b), 0.5 (c), 0.1 (d).
The value γ = 1 corresponds to the limit of a periodic chain.
Varying γ from one, we introduce quasiperiodicity (the most
pronounced for γ = 1/τ ≈ 0.618) in the chain. The γ = 1 case
recovers the periodic limit (with another period). The main
central sharp peak, corresponding to the periodic chain used
for (a), continuously transforms — through the set of selfsimilar patterns ((b) and (c)) corresponding to QP chains
(see Fig. 4) — to another peak (d) produced by a periodic
chain with the number of sites equal to the number of long
segments of the initial periodic chain with γ = 1.0 shown in
(a).
peaks have smaller pentagons inside them, which are inverted. As seen in Fig. 4, each subsequent subdivision
leads to a subset of peaks similar to the entire set of
peaks.
This analysis clearly demonstrates the self-similarity of
the distribution function of the vortices pinned on a QP
1D array of pinning sites. Similarly to the observed be-
In the previous section we studied the pinning of vortices by 1D QP chains of pinning sites. In particular, we
showed how the quasiperiodicity manifests itself in the
(p)
critical depinning current, Jc ∼ Nv /Nv , when increasing the applied magnetic flux, Φ ∼ Nv . We found that
the positions of the peaks of the function Jc (Φ) are governed by “harmonics” of long and short periods of the
QP chain of pinning sites. Independently of the length
of the chain (for Np ≥ 21), the peaks form a QP sequence
including the Fibonacci sequence as a fundamental subset. This self-similarity effect is clearly displayed in the
Fourier-transform of the distribution function of the vortices on a 1D QP array of pinning centers. The evolution
of QP peaks, when gradually changing the “quasiperiodicity” parameter γ, has revealed a continuous transition
from a QP chain — through the set of QP states (most
pronounced for γ = 1/τ ≈ 0.618) — to another periodic
chain, γ = 0, with a longer period. This phenomenon has
been studied both in real space and in reciprocal k-space.
In the present section and in the next sections, we analyze vortex pinning by 2D QP arrays; in particular, by
an array of pinning sites placed in the nodes of a fivefold Penrose lattice. Before tackling the Penrose-lattice
pinning array, let us start with a simplified system which
one can call “2D-quasiperiodic” (2DQP) since it is a 2D
system periodic in one direction (x-direction) and QP
9
in the other direction (y-direction).
A.
2D-quasiperiodic triangular lattice of pinning
sites
In an infinitely long one-dimensional homogeneous superconductor without any pinning centers, vortices obviously are equidistantly distributed, forming a periodic chain. Similarly, as it is well-known, in a threedimensional superconductor (or in quasi-two-dimensional
slabs or films), vortices organize themselves in a periodic triangular lattice shown schematically in Fig. 6a.
If we keep the lattice undistorted along the x-direction
and introduce a quasiperiodic deformation along the ydirection, similarly to the case of a 1D QP chain, we
obtain a 2QP triangular lattice as shown in Figs. 6b,c.
The “quasiperiodicity” parameter γy for this 2DQP lattice is defined as the ratio of the short to long periods, aL to aS (see Fig. 6c). The 2DQP triangular arrays of pinning sites are shown
√ for γy = 1.0 (Fig. 6a),
γy = 1/τ, where τ = (1 + 5)/2 (Fig. 6b), γy = 0.5
(Fig. 6c). The corresponding functions Jc (Nv ∼ Φ) are
presented in Figs. 6d,e,f for the following pinning parameters: fp /f0 = 2.0, and rp = 0.1λ.
For the triangular array of pinning sites (Fig. 6a),
we obtain a well-resolved main commensurability peak
(Fig. 6d), corresponding to the first matching field, at
Φ = Φ1 . Note that the vortex lattice is in general incommensurate with a triangular lattice of pinning sites
for the second matching field, i.e. when Φ = 2Φ1 (see,
e.g., Ref. [25]). For instance, for the parameters used in
our simulations, only each second row of pinning sites is
occupied at Φ = 2Φ1 , resulting in a very weak maximum
of the function Jc (Nv ∼ Φ) at that point; the parameters used (fp /f0 = 2.0, rp = 0.1λ) are nearly optimal
for revealing features of the function Jc (Φ) related to
quasiperiodicity.
When tuning γ out of the periodic value γy = 1.0,
the main peak decreases in magnitude, and a maximum
forms near it at a larger value of Nv ∼ Φ. These changes
are demonstrated in Fig. 6e (the corresponding pinning
array is shown in Fig. 6b) for γy = 1/τ .
It should be noted that here the parameter γy has a
different meaning, in the case of triangular 2DQP lattices, compared to the case of γ for the 1D QP chains
considered in the previous section. In a 1D QP chain,
γ = aS /aL is the ratio of distances between pinning sites,
whereas in a triangular 2DQP lattice γy defines the ratio
of the distances between the rows of pinning sites (see
Fig. 6c). It is easy to show that the ratio of distances between the neighboring pinning sites in a triangular 2DQP
lattice is
s
(1 + γy2 )a2 + 12a2
γ′ =
,
(14)
(1 + γy2 )a2 + 12γy2 a2
where a is the period of the 2DQP triangular lattice along
the (periodic) x-direction. Thus, for γy = 1/τ , the parameter γ ′ defined by Eq. (14) becomes γ ′ ≈ 0.7.
For γy = 0.5 (γ ′ ≈ 0.6), the function Jc (Nv ) is plotted
in Fig. 6f. The main peak is further depressed, while
the closest satellite peak becomes more pronounced. In
addition, other satellite peaks appear, which are much
less pronounced.
VI.
TRANSITION FROM A TRIANGULAR TO
A QUASIPERIODIC PENROSE-LATTICE
ARRAY OF PINNING SITES
Above, we have revealed some features of the behavior
of the critical depinning current Jc as a function of the
applied magnetic flux Φ (or as a function of the number
of vortices in the system Nv ∼ Φ). They have been
found under the transformation of a triangular lattice to
a 2DQP triangular lattice (array) of pinning sites.
Consider now a similar procedure but the final configuration of the transformation from a triangular lattice will
be a 2D QP array of pinning sites, namely, an array of
pinning sites located at the nodes of a five-fold Penrose
lattice. This kind of lattice is representative of a class
of 2D QP structures, or quasicrystals, which are referred
to as Penrose tilings. These structures possess a local
order and a rotational (five- or ten-fold) symmetry, but
do not have translational long-range order. Being constructed of a series of building blocks of certain simple
shapes combined according to specific local rules, these
structures can extend to infinity without any defects [11].
Below, we will discuss in more detail the structure of the
Penrose lattice.
The transformation of a triangular lattice to the Penrose one is a rather non-trivial procedure, as distinct
from the transformation to a 2DQP triangular lattice
done above when we simply stretched some of the interrow distances and squeezed other ones according to the
Fibonacci rules (Eq. (8)) for a one-dimensional QP lattice. In order to find intermediate configurations between
the triangular lattice and the Penrose lattice, we employ
the following approach. First, we place non-interacting
vortices at the positions coinciding with the nodes of
the Penrose lattice (these can be considered as pinning
sites for vortices, which right afterwards are “switched
off”). Then we let the vortices freely relax undergoing
the vortex-vortex interaction force at low temperatures
(and no pinning force). The vortices relax to their ground
state, which is a triangular lattice. During the relaxation process, we do a series of “snapshots”, recording
the coordinates of the vortex configurations at different
times. The sets of coordinates obtained are then used
as coordinates of pinning sites. We arrange these sets
in antichronological order to model a continuous transition of the pinning array from its initial configuration, a
triangular lattice, to its final configuration, a Penrose
10
(a)
1.0
2
y (l)
g = 1.0
(d)
Jc ~ Nv(p)/Nv
4
0
-2
0.5
-4
-6
-4
-2
0
2
6
4
x (l)
Jc ~ Nv(p)/Nv
4
2
y (l)
g = 0.618
1.0
(b)
0
(e)
Periodic along x
PQ along y
0.5
-2
-4
-6
-4
-2
0
2
6
4
x (l)
g = 0.5
(c)
aL aS
4
Jc ~ Nv(p)/Nv
1.0
y (l)
2
0
(f)
Periodic along x
PQ along y
0.5
-2
-4
-6
-4
-2
0
x (l)
2
4
6
0
50
100
Nv
150
FIG. 6: The spatial distribution of pinning sites for a triangular lattice (i.e., periodic) (a), and 2D-quasiperiodic (2DQP)
triangular, i.e. periodic along one direction (the x-direction) and QP along the other one (the y-direction), (b) and (c). The
parameter γy is defined as a ratio of short (aS ) to long √
(aL ) periods in the y-direction, as shown in (c). The values of the
parameter γy are: γy = 1.0 (a), γy = 1/τ, where τ = (1 + 5)/2 (golden mean) (b), γy = 0.5 (c). The critical depinning current
Jc , as a function of the number of vortices, Nv ∼ Φ, for triangular (d) and the 2DQP triangular lattices [shown in (b) and (c)]
(e) and (f), correspondingly, for fp /f0 = 2.0, rp = 0.1λ.
11
FIG. 7: (This figure is available in “png” format: “Penrose
Fig 7.png”; color) Left column: Transformation of a triangular lattice of pinning sites to a five-fold Penrose lattice.
The distributions of the pinning sites in the triangular lattice
(shown by blue solid circles in (a) and by blue open circles
in (b), (c), and (d), for comparison). Intermediate configurations (shown by red solid circles in (b) and (c)) between
the triangular (a) and the Penrose lattice (shown by red solid
circles in (d)). Right column: The corresponding critical depinning currents, Jc , as a function of the applied magnetic
flux, Φ, for the pinning arrays shown in (a) to (d), respectively: for the triangular lattice (shown by blue solid line in
(e) and by blue dashed lines in (f), (g), and (h), for comparison); for the intermediate configurations (shown by red
solid lines in (f) and (g)); for the Penrose lattice (shown by
the red solid line in (h)). For the Penrose lattice case in (b),
the drop in Jc (Φ) is an artifact of the boundary conditions.
Namely, the Penrose lattices of pinning sites did not fit the
square cell used in the simulations. Thus, the freely-moving
vortices near the edges significantly decreased the value of Jc ,
especially near Φ1 . This problem will be dealt separately in
Fig. 14 and in Eq. (16).
lattice.
In Figs. 7a to 7d, four of these configurations of pinning sites are shown. The triangular lattice is presented
in Fig. 7a. Two intermediate configurations are shown
in Figs. 7b,c. The pinning sites plotted in Fig. 7d are
located on the vertices of a Penrose lattice. For comparison, a pinning array in the form of a triangular lattice
(Fig. 7a) is also presented in Figs. 7b,c,d as open blue
circles. The functions Jc (Φ) calculated for the pinning
arrays shown in Figs. 7a to d, are plotted, respectively,
in Figs. 7e to h. The function Jc (Φ) for the triangular lattice (Fig. 7e) is also plotted, for comparison, in
Figs. 7g,f,h as a blue dashed curve.
The main commensurability peak related to the first
matching field in a triangular lattice of pinning sites, observed at Φ = Φ1 (Fig. 7e), turns out to be rather stable with respect to moderate deformations of the lattice
(Fig. 7f, see also Fig. 7b). It still has a maximum height
in Fig. 7f, although it broadens. However, the depths of
the valleys near the peak decreases by about 20 to 30 per
cent. Two sharp peaks near Φ = Φ1 /3 and Φ = Φ1 /6
(Fig. 7e), related to the commensurability of the longrange order in a triangular lattice, disappear. With further deformation, e.g., for the pinning array shown in
Fig. 7c, the main peak still remains but only about 80
per cent of the vortices are pinned in this case. The function Jc (Φ) becomes somewhat smoother, and it does not
display any pronounced features (for Φ & 2Φ1 /3) except
the main maximum.
The transition to a Penrose-lattice array of pinning
sites (Fig. 7d) is accompanied by the appearance of a
specific fine structure of the function Jc (Φ). Namely,
two well-resolved features on the broad main maximum
(Fig. 7h) are the most pronounced ones. Other, less
pronounced, features will be discussed below for larger
Penrose-lattice arrays. For large arrays, the function
Jc (Φ) is much less affected by fluctuations related to
the entrance of each single vortex in the system, which
are significant for the small-size array shown in Fig. 7d
(Np = 56). This small-size array is used here just as an
illustration, for studying the transition form a periodic
(triangular) to a QP (Penrose lattice) pinning site array.
However, studying even a relatively small piece of a QP
structure provides some useful information about properties of the whole system based on the self-similarity of
the lattice, which was revealed for 1D QP chains in the
previous section, and which will be demonstrated below
for 2D QP structures.
The Penrose and the 2DQP triangular lattice (Fig. 6f)
both have an important similar feature: their Jc (Φ) has
two nearby maxima. Thus, our previous analysis based
on several alternative ways to continuously deform a periodic lattice to a QP one shows that the features shown
are hallmarks of QP pinning arrays.
In the next section, the origin of the features observed
will be explained on the basis of a detailed analysis of the
structure and of the building blocks forming a five-fold
Penrose lattice. Other, less pronounced, features will also
be discussed. Some of them will be found in larger arrays
in our numerical simulations.
VII.
ANALYSIS OF THE FINE STRUCTURES
OF THE FUNCTION Jc (Φ) IN A
QUASIPERIODIC PENROSE-LATTICE ARRAY
OF PINNING SITES
The structure of a five-fold Penrose lattice is shown in
Fig. 8. As an illustration, a five-fold symmetric fragment
which consists of 46 points (nodes of the lattice) is presented (Fig. 8a). According to specific rules, the points
are connected by lines in order to display the structure of
the Penrose lattice (compare, e.g., to Fig. 7d). The elemental building blocks are rhombuses with equal sides
a snd angles which are multiples of θ = 36o . There
are rhombuses of two kinds forming the Penrose lattice
(Fig. 8b): (i) those having angles 2θ and 3θ (so called
“thick”; they are empty in Fig. 8), and (ii) rhombuses
with angles θ and 4θ (so called “thin”; they are colored
in orange in Fig. 8).
Let us analyze the structure of the Penrose lattice from
the point of view ot its pinning properties, when pinning
sites are placed in the vertices of the lattice. In particular, we are interested whether any specific matching
effects can exist in this system between the pinning lattice and the interacting vortices, which define the critical
depinning current at different values of the applied magnetic field (i.e., the function Jc (Φ)).
On the one hand, QP (quasicrystalline) patterns are
intrinsically incommensurate with the flux lattice for any
value of the magnetic field [16, 17, 18, 19], therefore, in
contrast to periodic (e.g., triangular or square) pinning
12
(a)
(b)
“Thick” rhombus
a
1.618a
a
1.176a
“Thin” rhombus
a
1.902a
a
0.618a
FIG. 8: (Color) The structure of a five-fold Penrose lattice
(a). The elemental building blocks are rhombuses with equal
sides a and angles which are multiples of θ = 36o . There are
rhombuses of two kinds: those having angles 2θ and 3θ (so
called “thick”), and rhombuses with angles θ and 4θ (so called
“thin”) (b). The distances between the nodes, i.e. the lengths
of the diagonals of the rhombuses
√ are: 1.176a (the short diagonal of a thick rhombus); (1+ 5)a/2√= τ a ≈ 1.618a (the long
diagonal of a thick rhombus); [(1 + 5)/2 − 1]a = (τ − 1)a ≈
0.618a, (the short diagonal of a thin rhombus); 1.902a (the
short diagonal of a thin rhombus) (b).
arrays, one might a priori assume a lack of sharp peaks
in Jc (Φ) for QP arrays of pinning sites.
On the other hand, the existence of many periods in
the Penrose lattice can lead to a hierarchy of matching
effects for certain values of the applied magnetic field,
resulting in strikingly-broad shapes for Jc (Φ).
In order to match the vortex lattice on an entire QP
pinning array, the specific geometry of the elements which
form the QP lattice is important as well as their arrangement, as distinct from the flux quantization effects and
superconductor-to-normal phase boundaries for which
the areas of the elements only plays a role [17, 18, 19].
As mentioned above, a five-fold Penrose lattice is constructed of building blocks, or rhombuses, of two kinds.
While the sides of the rhombuses are equal (denoted by
a), the distances between the nodes (where we place pin-
ning sites) are not equal (which is problematic for vortices). The lengths of the diagonals of the rhombuses
are as follows (Fig. 8b):
√ 1.176a (the short diagonal of a
thick rhombus); (1 + 5)a/2 = τ a ≈ 1.618a, where τ is
the golden
mean (the long diagonal of a thick rhombus);
√
[(1+ 5)/2−1]a = (τ −1)a ≈ 0.618a, (the short diagonal
of a thin rhombus); 1.902a (the short diagonal of a thin
rhombus).
Based on this hierarchy of distances, we can predict
matching effects (and corresponding features of the function Jc (Φ)) for the Penrose-lattice pinning array.
First, we can expect that there is a “first matching
field” (let us denote the corresponding flux as Φ1 ) when
each pinning site is occupied by a vortex. Although sides
of all the rhombuses are equal to each other similarly to
that in a periodic lattice, nevertheless this matching effect is not expected to be accompanied by a sharp peak.
Instead, it is a broad maximum since it involves three
kinds of local “commensurability” effects of the flux lattice: with the rhombus side a; with the short diagonal
of a thick rhombus, 1.176a, which is close to a; and with
the short diagonal of a thin rhombus, which is the golden
mean times a, 0.618a (see Fig. 8b).
It should be noted that this kind of matching assumes
that a vortex lattice is rather weak, i.e. the effect can be
more or less pronounced depending on the specific relations between the vortex-vortex interaction constant and
the strength of the pinning sites as well as on the distance between pinning sites and their radius. Assuming
that the vortex-vortex interaction constant is a material
parameter, all others can be adjustable parameters in
experiments with artificially-created QP pinning arrays.
For instance, the pinning parameters can be “adjusted”
by using as pinning centers antidots, i.e. microholes of
different radii “drilled” in a superconductor film [1, 2, 6],
or blind antidots [7] of different depths and radii.
Further, we can deduce that next to the above “main”
matching flux there is another matching related with the
filling of all the pinning sites in the vertices of thick rhombuses and only three out of four of the pinning sites in the
vertices of thin rhombuses, i.e., one of the pinning sites in
the vertices of thin rhombuses is empty. For this value of
the flux, which is lower than Φ1 , matching conditions are
fulfilled for two close distances, a (the side of a rhombus)
and 1.176a (the short diagonal of a thick rhombus) but
are not fulfilled for the short diagonal, a/τ , of the thin
rhombus.
Therefore, this QP feature is related to the golden
mean value, although not in such a direct way as in the
case of a 1D QP pinning array. This 2D QP matching results in a very wide maximum of the function Jc (Φ). The
position of this broad maximum, i.e., the specific value
of Φ (denoted here by Φvacancy/thin ≡ Φv/t = 0.757 Φ1)
could be found as follows. The ratio of the numbers
of thick and thin rhombuses is determined by the Fibonacci numbers and in the limit of large pinning arrays,
Np → ∞ this ratio tends to the golden mean. The number of unoccupied pinning sites is governed by the num-
13
FIG. 9: (This figure is available in “png” format: “Penrose Fig 9.png”; color) The critical depinning current Jc as a
function of the applied magnetic flux, Φ ∼ Nv , for an array
of pinning sites placed at the nodes of a five-fold Penrose lattice (for a part of the lattice which contains Np = 46 pinning
sites) (a). The distributions of vortices (shown by green dots)
pinned on the Penrose-lattice pinning site array (pinning sites
are shown by red open circles connected by orange solid lines
used in order to show the Penrose lattice structure, i.e., thick
and thin rhombuses), for specific values of the applied magnetic flux: (b) Φ = Φvacancy/thin ≡ Φv/t = 0.757 Φ1 , vortices
occupy all the pinning sites except those in one of the two vertices (connected by the short diagonal) of each thin rhombus,
each single and each pair of double thin rhombuses contain
one unoccupied pinning site at the matching field Φv/t ; (c)
Φ = Φ1 , the number of vortices Nv coincides with the number of pinning sites Np , and almost all the vortices are pinned
(because of using a square simulation cell, some of the vortices
are always “interstitial” but allow to keep the average vortex
density in the entire simulation cell; due to these additional
vortices, the value of the function Jc (Φ) effectively reduces by
a “filling” factor η = AP /A ≈ 0.575, where AP and A are the
areas of the Penrose-lattice “sample” and of the simulation
region; (d) Φ = Φinterstitial/thick ≡ Φi/T = 1.482 Φ1 , vortices
occupy all the pinning sites and interstitial positions inside
each thick rhombus, one vortex per each thick rhombus. The
parameters are fp /f0 = 2.0, rp = 0.1λ.
ber of thin rhombuses. However, some of the thin rhombuses are separated from other thin rhombuses by thick
ones (call them single thin rhombuses), but some of them
have common sides with each other (double thin rhombuses). Therefore, the number of vacancies (i.e., unoccupied pins) is then the number of single thin rhombuses
plus one half of the number of “double” thin rhombuses,
1 d
s
Npun (Φv/t ) = Nrh
+ Nrh
,
2
(15)
where Npun is the number of unoccupied pinning sites at
s
d
Φ = Φv/t , Nrh
and Nrh
are the numbers of single and
double thin rhombuses, correspondingly.
For higher vortex densities (e.g., for Φ =
Φinterstitial/thick ≡ Φi/T = 1.482 Φ1), we can expect
the appearance of a feature (maximum) of the function Jc (Φ) related to the entry of a single interstitial
vortex into each thick rhombus. The position of this
maximum is determined by the number of vortices at
Φ = Φ1 , which is Nv (Φ) = Np , plus the number of
thick
thick rhombuses, Nrh
. Since the ratio of the number
of thick to
that
of
thin
rhombuses is the golden mean,
√
τ = (1 + 5)/2 (in an infinite lattice; in a finite lattice,
it is determined by a ratio of two successive Fibonacci
thick
numbers), then Nrh
= Nrh /τ , where Nrh is the total
number of rhombuses. Here we used: 1/τ = τ − 1.
In Fig. 9a, the function Jc (Φ) is plotted for an array of
pinning sites in the form of a part of the Penrose lattice,
thick
shown in Figs. 9c,d,e, which consists of 20 thick (Nrh
=
thin
= 15) rhombuses and contains 46
20) and 15 thin (Nrh
nodes (pinning sites). The nodes are connected by lines
in order to show the rhombuses.
The distribution of vortices for Φ = Φ1 is shown in
Fig. 9c. The number of vortices Nv coincides with the
number of pinning sites Np , and almost all the vortices
are pinned. Note that since we use a square simulation
cell, some of the vortices are always outside the “Penrose
sample”. These vortices mimic the externally applied
magnetic field and determine the average vortex density
in the entire simulation cell. Because of these additional
vortices, the value of the function Jc (Φ) effectively reduces approximately by a “filling” factor η which is
η=
AP
≈ 0.575.
A
(16)
Here AP and A are the areas of the Penrose lattice (i.e.,
the area of all the rhombuses) and of the simulation region.
The value of the function Jc (Φ) in the maximum
Φ = Φ1 (Fig. 9a) is Jc ≈ 0.55, i.e. corresponds to almost perfect matching (two pinning sites occurred to be
unoccupied in the distribution shown in Fig. 9c) taking
into account Eq. (16).
Let us now more carefully analyze the calculations of
Jc (Φ) for the Penrose lattice, presented in Fig. 9. In
Fig. 9b, the distribution of vortices is shown for Φ =
Φv/t . Vortices occupy all the pinning sites except those
situated in one of the two vertices, connected by the short
diagonal, of each thin rhombus. Thus, each single and
each pair of double thin rhombuses contain one vacancy
(unoccupied pinning site) at the matching field Φv/t . The
corresponding maximum is indicated by the arrow (b) in
Fig. 9a.
The location of vortices for Φ = Φ1 (the maximum
(c) in Fig. 9a) is shown in Fig. 9c; here the number of
vortices Nv coincides with the number of pinning sites
Np , and almost all the vortices are pinned.
The distribution of vortices for Φ = Φi/T (Fig. 9d) is
also in agreement with our expectation: vortices occupy
all the pinning sites (there is only a single “defect” in
the distribution shown in Fig. 9d: one vortex left the
pinning site and became interstitial) plus interstitial positions inside each thick rhombus, i.e., one vortex per
each thick rhombus. However, the corresponding feature
of the function Jc (Φ) (arrow (d) in Fig. 9a) is less pronounced than the two above maxima at Φ = Φ1 and at
Φ = Φv/t .
In addition, there is a weak feature of the function
Jc (Φ) at Φ ≈ Φv/t /2, which more clearly manifests itself
for larger Penrose-lattice pinning arrays (see Fig. 10a).
Therefore, the calculated distributions of the vortices
pinned on the Penrose-lattice pinning site array and the
resulting function Jc (Φ) have revealed the QP features
which are in agreement with our expectations. The specific structure of the function Jc (Φ) is consistent with
two previous derivations both based on continuously deforming a QP lattice into a Penrose one (Sections V and
14
VI).
In Fig. 10a, the function Jc (Nv ) is shown for a larger
Penrose-lattice array of pinning sites, Np = 301. The
above QP features in Jc (Nv ) are much more pronounced
in this case then for smaller arrays because of a considerable reduction of the “noise” related with an entry of
each single vortex in the system.
In particular, the main maximum of the function
Jc (Nv ), which corresponds to the matching condition
Φ = Φ1 (Nv = 301), transforms into a rather sharp peak
with the magnitude η. Also, a local maximum of Jc (Φ)
at Φ ≈ Φv/t /2, is more pronounced for Np = 301, as
mentioned above.
Finally, Fig. 11 demonstrates the function Jc (Φ), calculated for different samples with Np = 46, 141, and
301 (Fig. 11a), and also for different criteria of Jc : for
the “static” and dynamical criteria (Fig. 11b). In the dynamical simulations of Jc using a threshold criterion, i.e.,
Jc is obtained as the minimum current J ∝ fid which
depins the vortices. In the Appendix A, we show onset of vortex motion when the applied current J exceeds
the critical currnet: J > Jc . The results obtained using
these two criteria are essentially equal, and throughout
this work we use the “static” criterion defined above.
VIII.
ANALYTICAL APPROACH
The following analysis could provide a better understanding of the above structure of Jc (Φ) for the Penrose
pinning lattice. Let us compare the elastic Eel and pinning Epin energies of the vortex lattice at H1 and at (the
lower field) Hv/t , corresponding to the two maxima of Jc
(e.g., Figs. 9a, 10a, 11). Vortices can be pinned if the
gain Epin = Upin β npin of the pinning energy is larger
than the increase of the elastic energy [31, 32, 33] related
to local compressions:
Eel = C11
(aeq − b)2
.
aeq
(17)
The shear elastic energy (∝ C66 ) provides the same qualitative result. Here, Upin ∼ fp rp , npin is the density of
pinning centers,
β(H ≤ H1 ) = H/H1 = B/(Φ0 npin ),
1.0
(a)
Penrose, Np = 46
Penrose, Np = 141
Penrose, Np = 301
Nvp/Nv
0.8
0.6
0.4
0.2
F1
0.0
1.0
Jc ~ Nvp/Nv , Jc /Jc(Nv = 1)
FIG. 10: (This figure is available in “png” format: “Penrose
Fig 10.png”; color) The critical depinning current Jc as a
function of the applied magnetic flux, Φ ∼ Nv , for an array of
pinning sites placed in the nodes of a five-fold Penrose lattice
(for Np = 301 ) (a). The distributions of vortices (shown
by green solid circles) pinned on the Penrose-lattice pinning
site array (shown by red open circles), for specific values of
the applied magnetic flux which correspond to two matching
fields: (b) Φ = Φv/t , vortices occupy all the pinning sites
except one in each thin rhombus; (c) Φ = Φ1 , all the vortices
are pinned. The parameters are the same as in Fig. 9.
(b)
0.8
F
Penrose, Np = 141
Jc ~ Nvp/Nv (”Static” Criterion)
Jc /Jc(Nv = 1) (Dynamic Criterion)
0.6
0.4
0.2
0
0.5
1
F/F1
FIG. 11: (Color) (a) The critical depinning current Jc as a
(p)
function of the applied magnetic flux, Φ ∼ Nv /Nv , shown
for three different Penrose-lattice pinning site arrays: Np =
46 (shown by blue solid line), Np = 141 (shown by green
solid line), Np = 301 (shown by red solid line). The revealed
features are hallmarks of a Penrose-lattice pinning site array.
(b) Comparison of Jc as a function of the applied magnetic
flux, calculated using the “static” and dynamical criteria. The
latter means calculating Jc using a threshold criterion, i.e., Jc
is obtained as the minimum current J ∝ fid which depins
the vortices. The results obtained using these two criteria are
essentially equal.
and β(H > H1 ) = 1 is the fraction of occupied pinning
sites (β = 1 for H = H1 , and β = 0.757 for H = Hv/t ),
√
1/2
aeq = 2/ 3βnpin
is the equilibrium distance between vortices in the triangular lattice, b is the minimum
distance between vortices in the distorted pinned vortex
lattice (b = a/τ for H = H1 and b = a for H = Hv/t ),
and
C11 =
B2
4π(1 + λ2 k 2 )
(18)
is the compressibility modulus for short-range deforma1/2
tions [31] with characteristic spatial scale k ≈ (npin ) .
15
Epin − Eel =
βfdiff npin Φ20
,
4πλ2
(19)
where
fdiff =

4πλ2 Upin
− β 1 − b
Φ20
!1/2 2
√
β 3npin
 .
2
1.0
Jc ~ Nv(p)/Nv
The dimensionless difference of the pinning and elastic
energies is
fdiff µ Epin - Eel
No
peak
Broad
peak
Two
peaks
2
4pUpinl /F0
0.6
Strong
pinning
(20)
The function fdiff 0 is shown schematically in the inset
to Fig. 12. Near matching fields, Jc has a peak when
fdiff > 0 (and no peak when fdiff < 0). Since only two
matching fields provide fdiff > 0, then our analysis explains the two-peak structure observed in Jc shown in
Figs. 9a, 10a, 11. For instance, for the main matching
fields Eq. (20) gives: fdiff (Φv/t ) ≈ 0.0056, fdiff (Φ1 ) ≈
0.0058, and fdiff (Φi/T ) ≈ −0.09. Note that for weaker
pinning, the two-peak structure gradually turns into one
very broad peak, and eventually zero peaks for weak
enough pinning (see Fig. 12). The Jc peaks corresponding to higher matching fields are strongly suppressed because of the fast increase (∝ B 2 ) of the compressibility modulus C11 and, thus, the elastic energy with respect to the pinning energy; the latter cannot exceed the
maximum value Upin npin . The subharmonic peaks of Jc ,
which could occur for lower fields H < Hv/t , are also
suppressed due to the increase of C11 associated with the
growing spatial scales 1/k of the deformations.
IX. THE CRITICAL CURRENT Jc (Φ) IN A
RANDOMLY DISTORTED TRIANGULAR
LATTICE
Above we have studied the function Jc (Φ) for periodic,
QP 2D arrays of pinning sites and analyzed the transition
from the periodic triangular lattice to the QP Penrose
lattice (see Fig. 7). One of the issues, which is related
to this analysis and can be useful for practical applications, is the increase of the critical current (shown, e.g.,
in Fig. 7f) in the regions corresponding to minima of
Jc (Φ) for periodic (triangular) pinning arrays. The situation shown in Fig. 7f seems to be the optimal from the
point of view of a homogeneous increase of Jc (Φ) without degradation of the main peak at Φ = Φ1 . Recall that
it corresponds to a slightly “quasiperiodically distorted”
triangular lattice (see Fig. 7b). The pinning sites of the
triangular lattice are shifted from their “correct” positions but not randomly: their positions are determined
by vortex-vortex interaction, which tries to restore the
triangular lattice, and by the memory about previous
configurations including the initial one, i.e., the Penrose
lattice. Analyzing the lattice presented in Fig. 7b we can
deduce that it keeps some short-range order (i.e., distorted triangular cells similar in shape and size to those
in the triangular lattice) but does not have long-range order of the triangular lattice. As a result, the main peak
0.4
fp/f0 = 2.0
0.2
fp/f0 = 1.4
fp/f0 = 0.5
0.0
Weak
pinning
Very
weak pinning
F1
F
FIG. 12: (Color) The critical current Jc (Φ) for Penroselattice arrays for different pinning strength fp /f0 = 2.0 (red
solid line), fp /f0 = 1.4 (green solid line), fp /f0 = 0.5 (blue
solid line). The peak at Φ1 is suppressed for weaker pinning
(fp /f0 = 1.4). Eventually, all the main peaks disappear for
sufficiently weak pins (fp /f0 = 0.5). The inset shows the
dimensionless difference, fdiff , of the pinning and the elastic
energies versus the pinning-to-interaction energy ratio, for the
broad Jc peak at Φv/t (red dashed line) and for Φ1 (red solid
line). Only fdiff > 0 gives stable peaks in Jc .
remains, since it is related to the short-order matching
effects (i.e., over the distances of the order inter-site spacings a). The sharp decrease around the maximum and
appearance of the deep valleys is explained by the absence (due to long-range order, i.e., over distances longer
than a) of any matching effects for the flux densities
close to Φ = Φ1 (there is no matching for Φ = Φ1 /2 or
Φ = 1.5Φ1 and Φ = 2Φ1 for the triangular lattice). When
the long-range order is destroyed, as shown in Fig. 7b,
matching effects other than Φ = Φ1 become allowed.
It is appropriate to mention here that the QP Penrose lattice possesses a short-range order but does not
have a long-range translational order. In such a way, the
“quasiperiodically distorted” triangular lattice (Fig. 7b)
or the QP Penrose lattice itself are good candidates for
the optimal enhancement of the critical current in the
regions where the function Jc (Φ) have minima for a periodic lattice. (It should be noted here that the curves
shown, e.g., in Fig. 7h, for the triangular and the Penrose lattices are calculated for the same cell, although
the effective area of the Penrose lattice is smaller than
that of the triangular lattice with the same number of
pinning sites. This discrepancy is taken into account by
the “filling factor” introduced by Eq. (16). It should be
also recalled, when comparing the function Jc (Φ) for the
case of the triangular and the Penrose lattices, that the
main maximum of the curve for the Penrose lattice (see
Fig. 11) is the second sharp peak at Φ = Φ1 .)
In this respect, it is interesting to compare the above
16
FIG. 13: (This figure is available in “png” format: “Penrose
Fig 13.png”; color) Left column: Randomly distorted triangumax
lar lattices for rran
= 0.2(a/2) (a), 0.3(a/2) (b), 0.4(a/2) (c),
and for a/2 (d). For comparison, the triangular lattice is also
shown. Right column: The functions Jc (Φ) corresponding to
the distributions of pinning sites shown in (a) to (d), are presented, respectively, in (e) to (h). At low levels of noise (e),
(f), the valleys (minima) start to fill due to disappearance of
the long-range order, similarly to the case of the Penrose lattice, although accompanied with a weaker enhancement of Jc
than for the case of the “quasiperiodic distortion” (Fig. 7f).
For higher levels of noise, the main peak degrades without
any essential enhancement of Jc in the neighborhood (g), (h).
a weaker enhancement of Jc than for the case of the
“quasiperiodic distortion” (Fig. 7f). For higher levels
of noise, the main peak degrades without any essential
enhancement of Jc in the neighborhood (Figs. 13g, h).
For comparison, we also show in Fig. 14 the Jc (Φ) for a
Penrose-lattice (calculated for the sample with Np = 301,
only for the area of the Penrose lattice AP , see Eq. (16)).
Notice that the QP lattice leads to a very broad and potentially useful enhancement of the critical current Jc (Φ),
even compared to the triangular or random APS. The remarkably broad maximum in Jc (Φ) is due to the fact that
the Penrose lattice has many (infinite, in the thermodynamic limit) periodicities built in it [11]. In principle,
each one of these periods provides a peak in Jc (Φ). In
practice, like in quasicrystalline difraction patterns, only
few peaks sre strong. This is also consistent with our
study. Furthermore, the pinning parameters can be adjusted by using as pinning centers either antidots (microholes of different radii “drilled” in the film [1, 6], or
blind antidots [7] of different depths and radii. Thus, our
results could be observed experimentally.
X.
FIG. 14: (Color) The critical current Jc (Φ) for a 301-sites
Penrose-lattice (dark green solid line), (recalculated for flux
only on the Penrose area, AP ), triangular (black solid line)
and random (red solid line) pinning arrays. The Penrose lattice provides a remarkable enhancement of Jc (Φ) over a very
wide range of values of Φ because it contains many periods in
it.
results for the “quasiperiodic distortion” of the triangular
lattice with its random distortion. For this purpose, we
introduce a random angle αran : 0 < αran < 2π, and a
max
random radius of the displacement dran : 0 < rran < rran
,
max
where rran is the maximal displacement radius, which is
a measure of noise measured in units of a/2, where a is
the (triangular) lattice constant.
In Fig. 13, randomly distorted triangular lattices are
max
shown for rran
= 0.2(a/2) (Fig. 13a), 0.3(a/2) (Fig. 13b),
0.4(a/2) (Fig. 13c), and for a/2 (Fig. 13d). For comparison, the triangular lattice is also shown in Figs. 14a to
d. The corresponding functions Jc (Φ) are presented, respectively, in Figs. 13e to h. At low levels of noise (e.g.,
Fig. 13e, f) the valleys (minima) start to fill due to the
disappearance of the long-range order, similarly to the
case of the Penrose lattice, although accompanied with
CONCLUSIONS
The critical depinning current Jc , as a function of the
applied magnetic flux Φ, has been studied in QP pinning
arrays, from one-dimensional chains to two-dimensional
arrays of pinning sites set in the nodes of quasiperiodic
lattices including a 2D-quasiperiodic triangular lattice
and a five-fold Penrose lattice.
In a 1D quasiperiodic chain of pinning sites, positions of the peaks of the function Jc (Φ) are governed by
“harmonics” of long and short periods of the quasiperiodic chain. Independently of the length of the chain,
the peaks form a set of quasiperiodic sequence including a Fibonacci sequence as a basic subset. Analyzing
the evolution of the peaks, when a continuous transition is performed from a periodic to a quasiperiodic lattice of the pinning sites, we found that the peaks related to the Fibonacci sequence are most pronounced
when the ratio of lengths of the long and the short periods is the golden mean. A comparison of the sets
of peaks for different chains shows that the functions
Jc (Φ) for the 1D quasiperiodic chain is self-similar. In
the k-space, the self-similarity effect is displayed in the
Fourier-transform of the distribution function of the system of vortices pinned on a 1D quasiperiodic array of pinning centers. The evolution of quasiperiodic peaks when
gradually changing the “quasiperiodicity” parameter γ
(i.e., ratio of the lengths of short to long elements of a
quasiperiodic chain) has revealed a continuous transition
from a periodic chain – through the set of quasiperiodic
states – to another periodic chain with a longer period.
This phenomena has been studied both in real space and
in reciprocal k-space.
In 2D quasiperiodic pinning arrays (e.g., Penrose lattice), the pinning of vortices is related to matching
17
conditions between a triangular vortex lattice and the
quasiperiodic lattice of the pinning centers. Although
more complicated than in 1D pinning chains, the specific
behavior of Jc (Φ) is determined by the presence of two
different kinds of elements – thick and thin rhombuses –
forming the quasiperiodic lattice. Based on these considerations, the positions of the main maxima of Jc (Φ) for
Penrose lattice are predicted.
In particular, for the first matching field each pinning
site is occupied by a vortex. The corresponding maximum of the function Jc (Φ) is broad since it involves at
least three kinds of local matching effects of the flux lattice, with the rhombus side and with short diagonals of
thick and thin rhombuses.
Another Penrose-lattice matching field is related with
local matching effects which involve the intervortex distance of the vortex flux lattice, the rhombus side and the
short diagonal of thick rhombus. For this field, all the
pinning sites are occupied, which are situated in the vertices of thick rhombuses and only three out of four in the
vertices of thin rhombuses. The number of unoccupied
pinning sites is governed by the number of thin rhombuses. Some of the thin rhombuses are single (i.e., separated from other thin rhombuses by thick ones), while
some of them are double (i.e., have common sides with
each other). Therefore, the number of vacancies is the
number of single thin rhombuses plus one half of the
number of “double” thin rhombuses. One more important feature of the function Jc (Φ) occurs for higher vortex densities, when a single interstitial vortex enters each
thick rhombus.
Numerical simulations performed for various sample
sizes have revealed a good agreement with our predictions.
The revealed features can be more or less pronounced
depending on specific relations between the vortex-vortex
interaction constant and the strength of the pinning sites,
as well as on the distance between pinning sites and their
radius. While the vortex-vortex interaction constant is a
material parameter, all others can be adjustable parameters in experiments with artificially created quasiperiodic pinning arrays. This can be reached by using, for
instance, antidots (i.e., microholes “drilled” in a superconductor film) or blind antidots of different depths and
radii as pinning centers. Our calculations provide the
necessary relations between these parameters for possible experimental realizations.
A continuous deformation of the Penrose lattice to a
periodic triangular lattice (i) shows that the above revealed features are hallmarks of quasiperiodic pinning arrays; (ii) provides us with a tool for the controlled change
of the magnitude, sharpness and the position of the peaks
of Jc (Φ) that is important for possible applications. In
particular, our analysis shows that the quasiperiodic lattice provides an unusually broad critical current Jc (Φ),
that could be useful for practical applications demanding
high Jc ’s over a wide range of fields.
ACKNOWLEDGMENT
This work was supported in part by the National Security Agency (NSA) and Advanced Research and Development Activity (ARDA) under Air Force Office of
Scientific Research (AFOSR) contract number F4962002-1-0334; and also supported by the US National Science Foundation grant No. EIA-0130383, and RIKEN’s
President’s funds.
APPENDIX A: Onset of vortex motion for currents
higher then the critical current: J > Jc
Here we present vortex flow patterns for currents exceeding the critical value, J > Jc . In Fig. 15, the vortex
flow patterns are shown calculated for a Penrose sample
with Np = 141 (pinning sites shown by red circles). The
ground-state vortex configuration is shown in Fig. 15a
(vortices shown by green dots) for Φ ≈ Φ1 and when
no driving force is applied. This vortex configuration is
similar to those shown in Figs. 9c and 10c, when the number of vortices (within the sample area) is equal to the
number of pinning sites, and all the vortices are pinned.
When an increasing driving force fd ∼ J is applied, the
vortices do not move until fd reaches some threshold
value, when vortices depin. The current which corresponds to the driving force depinning the vortices, is then
defined as the critical current Jc (dynamical criterion). A
comparison of Jc ’s calculated using this criterion and the
static criterion, is shown above in Fig. 11b. When unpinned, the vortices move along complicated trajectories,
or “channels” created by pinning arrays and interacting
with other vortices. Figs. 15b to 15d show the onset of
the flux motion for J > Jc , following the traces of moving vortices over distances about 0.5λ (b), 1λ (c), and
2λ (d). On the subsequent consecutive snapshots, vortices are shown by a sequence of consecutive open blue
circles (and by blue solid circles for the last snapshot on
each panel). These show dynamical configurations of the
vortex lattice in motion. Note the appearance of local
“rivers” of vortices moving along the channels between
neighboring pinning sites.
18
FIG. 15: (This figure is available in “png” format: “Penrose
Fig 15.png”; color) Vortex flow patterns for J > Jc , calculated
for a Penrose sample with Np = 141 (pinning sites shown by
red circles). (a) Ground-state vortex configuration (vortices
shown by green dots) when Φ ≈ Φ1 , and no driving force
is applied. (b-d) The onset of the flux motion for J > Jc ,
following the traces of moving vortices over distances about
0.5λ (b), 1λ (c), and 2λ (d). On the subsequent consecutive
snapshots, vortex trajectories are shown by black dotted lines
(the blue solid circles show the last snapshot on each panel).
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This figure "Penrose_Fig_7.png" is available in "png" format from:
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